Subgroups — C₂×C₁₃×#2327 #15720407

Order 3328.

2958 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₁₃	C₂	1	2	yes	maximal
3	C₁₃×#2327	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₁₃×#391	C₂	1	2	yes	maximal
5	C₂×C₁₃×#387	C₂	1	8	yes	maximal
6	C₂×C₁₃×#382	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#376	C₂	1	1	yes	maximal, characteristic
8	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
9	C₂×C₁₃×C₄⋊Q₈	C₄	1	16	yes	complemented
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	8	yes	—
11	C₂×C₄×Q₈×C₁₃	—	2	16	no	—
12	C₂×C₄×Q₈×C₁₃	C₂²	1	16	yes	—
13	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
15	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
16	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₁₃	C₂²	1	4	yes	—
18	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#391	C₂²	1	16	yes	—
21	C₁₃×#387	C₂²	1	64	yes	—
22	C₁₃×#382	C₂²	1	8	yes	—
23	C₁₃×#376	C₂²	1	8	yes	—
24	C₂×C₁₃×#95	C₂²	1	4	yes	—
25	C₂×C₁₃×#95	C₂²	1	4	yes	—
26	C₂×C₁₃×#95	C₂²	1	8	yes	—
27	C₂²×Q₈×C₁₃	—	2	8	no	—
28	C₂²×Q₈×C₁₃	—	2	8	no	—
29	C₂²×Q₈×C₁₃	C₂×C₄	1	16	yes	—
30	C₂²×Q₈×C₁₃	C₂³	1	8	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
35	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
36	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
37	C₁₃×C₄⋊Q₈	C₂×C₄	1	64	yes	complemented
38	C₄×Q₈×C₁₃	—	2	64	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	16	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
41	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	32	yes	—
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂³	1	16	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂³	1	16	yes	—
45	C₂×C₄²×C₁₃	—	2	16	no	—
46	C₂×C₄²×C₁₃	—	2	4	no	—
47	C₂×C₄²×C₁₃	C₂×C₄	1	8	yes	—
48	C₂×C₄²×C₁₃	D₄	1	8	yes	—
49	C₂×C₄²×C₁₃	C₂³	1	16	yes	—
50	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
51	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
52	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
53	C₁₃×#95	C₂³	1	16	yes	—
54	C₁₃×#95	C₂³	1	32	yes	—
55	C₁₃×#95	C₂³	1	16	yes	—
56	C₂×#2327	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
58	C₂×Q₈×C₁₃	—	2	64	no	—
59	C₂×Q₈×C₁₃	—	2	16	no	—
60	C₂×Q₈×C₁₃	—	2	16	no	—
61	C₂×Q₈×C₁₃	C₂²×C₄	1	32	yes	—
62	C₂²×C₄×C₁₃	—	2	8	no	—
63	C₂²×C₄×C₁₃	—	2	8	no	—
64	C₂²×C₄×C₁₃	—	2	8	no	—
65	C₂²×C₄×C₁₃	—	2	8	no	—
66	C₂²×C₄×C₁₃	—	2	8	no	—
67	C₂²×C₄×C₁₃	—	2	4	no	—
68	C₂²×C₄×C₁₃	—	2	16	no	—
69	C₂²×C₄×C₁₃	—	2	8	no	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	16	yes	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₁₃	#41	1	4	yes	—
76	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
77	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
78	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
79	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
80	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
81	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
82	C₁₃×C₄⋊C₄	—	2	32	no	—
83	C₁₃×C₄⋊C₄	—	2	16	no	—
84	C₁₃×C₄⋊C₄	C₂²×C₄	1	64	yes	—
85	C₄²×C₁₃	—	2	32	no	—
86	C₄²×C₁₃	—	2	8	no	—
87	C₄²×C₁₃	C₂²×C₄	1	16	yes	—
88	C₄²×C₁₃	C₂×D₄	1	16	yes	—
89	C₂²×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
90	C₂²×C₄×Q₈	C₂×C₁₃	1	2	yes	—
91	#2327	C₂×C₁₃	1	16	yes	direct product factor
92	C₂×#391	C₂×C₁₃	1	2	yes	—
93	C₂×#387	C₂×C₁₃	1	8	yes	—
94	C₂×#382	C₂×C₁₃	1	1	yes	characteristic
95	C₂×#376	C₂×C₁₃	1	1	yes	characteristic
96	C₂³×C₁₃	C₂³×C₄	1	2	yes	—
97	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
98	C₂³×C₁₃	C₂×#41	1	2	yes	—
99	C₂³×C₁₃	2₋¹⁺⁴	1	8	yes	—
100	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
101	Q₈×C₁₃	—	2	64	no	—
102	C₂×C₄×C₁₃	—	2	32	no	—
103	C₂×C₄×C₁₃	—	2	16	no	—
104	C₂×C₄×C₁₃	—	2	16	no	—
105	C₂×C₄×C₁₃	—	2	8	no	—
106	C₂×C₄×C₁₃	—	2	8	no	—
107	C₂×C₄×C₁₃	—	2	8	no	—
108	C₂×C₄×C₁₃	—	2	8	no	—
109	C₂×C₄×C₁₃	—	2	8	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	16	no	—
112	C₂×C₄×C₁₃	—	2	8	no	—
113	C₂×C₄×C₁₃	—	4	16	no	—
114	C₂×C₄×C₁₃	—	4	8	no	—
115	C₂×C₄×C₁₃	C₄×D₄	1	16	yes	—
116	C₂×C₄×C₁₃	C₂³×C₄	1	8	yes	—
117	C₂×C₄×C₁₃	C₂³×C₄	1	16	yes	—
118	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
119	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
120	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
121	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
122	C₂³×Q₈	C₂²×C₁₃	1	2	yes	—
123	C₂×C₄⋊Q₈	C₄×C₁₃	1	16	yes	complemented
124	C₂×C₄⋊Q₈	C₂²×C₁₃	1	8	yes	—
125	C₂×C₄×Q₈	—	2	16	no	—
126	C₂×C₄×Q₈	C₂²×C₁₃	1	16	yes	—
127	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
128	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
129	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
130	C₂²×C₄²	C₂²×C₁₃	1	4	yes	—
131	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
132	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
133	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
134	#391	C₂²×C₁₃	1	16	yes	—
135	#387	C₂²×C₁₃	1	64	yes	—
136	#382	C₂²×C₁₃	1	8	yes	—
137	#376	C₂²×C₁₃	1	8	yes	—
138	C₂×#95	C₂²×C₁₃	1	4	yes	—
139	C₂×#95	C₂²×C₁₃	1	4	yes	—
140	C₂×#95	C₂²×C₁₃	1	8	yes	—
141	C₂²×C₁₃	C₂×C₄×D₄	1	4	yes	—
142	C₂²×C₁₃	#519	1	8	yes	—
143	C₂²×C₁₃	#536	1	8	yes	—
144	C₂²×C₁₃	#541	1	8	yes	—
145	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
146	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
147	C₂²×C₁₃	C₂²×#41	1	1	yes	characteristic
148	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	4	yes	—
149	C₄×C₁₃	—	2	16	no	—
150	C₄×C₁₃	—	2	8	no	—
151	C₄×C₁₃	—	2	8	no	—
152	C₄×C₁₃	—	4	8	no	—
153	C₄×C₁₃	—	4	4	no	—
154	C₄×C₁₃	C₂×C₄×D₄	1	8	yes	—
155	C₂²×Q₈	—	2	8	no	—
156	C₂²×Q₈	—	2	8	no	—
157	C₂²×Q₈	C₂×C₄×C₁₃	1	16	yes	—
158	C₂²×Q₈	C₂³×C₁₃	1	8	yes	—
159	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
160	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
161	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
162	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
163	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
164	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
165	C₄⋊Q₈	C₂×C₄×C₁₃	1	64	yes	complemented
166	C₄×Q₈	—	2	64	no	—
167	C₂×C₄⋊C₄	—	2	16	no	—
168	C₂×C₄⋊C₄	—	2	8	no	—
169	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	32	yes	—
170	C₂×C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
171	C₂×C₄⋊C₄	C₂³×C₁₃	1	16	yes	—
172	C₂×C₄⋊C₄	C₂³×C₁₃	1	16	yes	—
173	C₂×C₄²	—	2	16	no	—
174	C₂×C₄²	—	2	4	no	—
175	C₂×C₄²	C₂×C₄×C₁₃	1	8	yes	—
176	C₂×C₄²	D₄×C₁₃	1	8	yes	—
177	C₂×C₄²	C₂³×C₁₃	1	16	yes	—
178	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
179	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
180	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
181	#95	C₂³×C₁₃	1	16	yes	—
182	#95	C₂³×C₁₃	1	16	yes	—
183	#95	C₂³×C₁₃	1	32	yes	—
184	C₂×C₁₃	#2327	1	8	yes	direct product factor
185	C₂×C₁₃	C₂²×C₄×D₄	1	1	yes	characteristic
186	C₂×C₁₃	C₂×#519	1	2	yes	—
187	C₂×C₁₃	C₂×#536	1	2	yes	—
188	C₂×C₁₃	C₂×#541	1	2	yes	—
189	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
190	C₂×Q₈	—	2	64	no	—
191	C₂×Q₈	—	2	16	no	—
192	C₂×Q₈	—	2	16	no	—
193	C₂×Q₈	C₂²×C₄×C₁₃	1	32	yes	—
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	16	no	—
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	C₂²×C₄×C₁₃	1	8	yes	—
203	C₂²×C₄	C₂²×C₄×C₁₃	1	16	yes	—
204	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
205	C₂²×C₄	C₂×D₄×C₁₃	1	8	yes	—
206	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
207	C₂²×C₄	C₁₃×#41	1	4	yes	—
208	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
209	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
210	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
211	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
212	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
213	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
214	C₄⋊C₄	—	2	32	no	—
215	C₄⋊C₄	—	2	16	no	—
216	C₄⋊C₄	C₂²×C₄×C₁₃	1	64	yes	—
217	C₄²	—	2	32	no	—
218	C₄²	—	2	8	no	—
219	C₄²	C₂²×C₄×C₁₃	1	16	yes	—
220	C₄²	C₂×D₄×C₁₃	1	16	yes	—
221	C₁₃	C₂×#2327	1	1	yes	characteristic, direct product factor, Sylow
222	C₂³	C₂³×C₄×C₁₃	1	2	yes	—
223	C₂³	C₂²×D₄×C₁₃	1	2	yes	—
224	C₂³	C₂×C₁₃×#41	1	2	yes	—
225	C₂³	C₁₃×2₋¹⁺⁴	1	8	yes	—
226	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
227	Q₈	—	2	64	no	—
228	C₂×C₄	—	2	32	no	—
229	C₂×C₄	—	2	16	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	16	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	4	16	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	C₄×D₄×C₁₃	1	16	yes	—
242	C₂×C₄	C₂³×C₄×C₁₃	1	8	yes	—
243	C₂×C₄	C₂³×C₄×C₁₃	1	16	yes	—
244	C₂×C₄	C₂³×C₄×C₁₃	1	4	yes	—
245	C₂×C₄	C₂²×D₄×C₁₃	1	8	yes	—
246	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
247	C₂×C₄	C₂×C₁₃×#41	1	4	yes	—
248	C₂²	C₂×C₄×D₄×C₁₃	1	4	yes	—
249	C₂²	C₁₃×#519	1	8	yes	—
250	C₂²	C₁₃×#536	1	8	yes	—
251	C₂²	C₁₃×#541	1	8	yes	—
252	C₂²	C₂⁴×C₄×C₁₃	1	1	yes	characteristic
253	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
254	C₂²	C₂²×C₁₃×#41	1	1	yes	characteristic
255	C₂²	C₂×C₁₃×2₋¹⁺⁴	1	4	yes	—
256	C₄	—	2	16	no	—
257	C₄	—	2	8	no	—
258	C₄	—	2	8	no	—
259	C₄	—	4	8	no	—
260	C₄	—	4	4	no	—
261	C₄	C₂×C₄×D₄×C₁₃	1	8	yes	—
262	C₂	C₁₃×#2327	1	8	yes	direct product factor
263	C₂	C₂²×C₄×D₄×C₁₃	1	1	yes	characteristic
264	C₂	C₂×C₁₃×#519	1	2	yes	—
265	C₂	C₂×C₁₃×#536	1	2	yes	—
266	C₂	C₂×C₁₃×#541	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₁₃
Frattini, cosocle	C₂³	C₂⁵×C₁₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₁₃×#2327	C₂	direct (N × H)
9. C₂×C₁₃×C₄⋊Q₈	C₄	semidirect
37. C₁₃×C₄⋊Q₈	C₂×C₄	semidirect
56. C₂×#2327	C₁₃	direct (N × H)
91. #2327	C₂×C₁₃	direct (N × H)
123. C₂×C₄⋊Q₈	C₄×C₁₃	semidirect
165. C₄⋊Q₈	C₂×C₄×C₁₃	semidirect
184. C₂×C₁₃	#2327	direct (N × H)
221. C₁₃	C₂×#2327	direct (N × H)
262. C₂	C₁₃×#2327	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₁₃×C₄⋊Q₈	2
C₂²×C₄×Q₈×C₁₃	2
C₁₃×#2327	2
C₂×C₁₃×#391	2
C₂×C₁₃×#387	2
C₂×C₁₃×#382	2
C₂×C₁₃×#376	2
C₂×#2327	13

Subgroups — C2×C13×#2327 #15720407

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2327 #15720407